Question

Two bars A and B made of different material have same length. The coefficient of expansion and Young's modulus of bar B is twice that of bar A. If the temperature of both bars is increased by the same amount while preventing any expansion, then the ratio of stress developed in bar A to that in bar B is

• A. $16$
• B. $8$
• C. $4$
• D. $2$

Hint:

Thermal stress depends directly on the product of $\alpha$ and $E$ ($\sigma \propto \alpha E$). Since Bar B has twice the value for both properties, it experiences $2 \times 2 = 4$ times more internal thermal stress than Bar A.

Answer 1

The Correct Option is C

Step 1: Pick simple numbers that fit the given ratios.
Instead of working symbolically, take \( \alpha_A = 1 \) and \( E_A = 1 \). Since bar B's expansion coefficient and modulus are both twice bar A's, \( \alpha_B = 2 \) and \( E_B = 2 \).

Step 2: Compute the thermal stress for each bar.
With expansion fully prevented, \( \sigma = \alpha E \Delta T \). Taking \( \Delta T = 1 \) for both bars, since it is the same for both anyway: \[ \sigma_A = 1 \times 1 \times 1 = 1, \qquad \sigma_B = 2 \times 2 \times 1 = 4 \]

Step 3: Take the ratio.
\[ \frac{\sigma_A}{\sigma_B}=\frac14 \quad\Rightarrow\quad \frac{\sigma_B}{\sigma_A} = 4 \] \[ \boxed{4} \] This matches option (C).